Pam  00:01

Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam, a former mimicker turned mather.

Kim  00:09

And I'm Kim, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching.

Pam  00:16

We know that algorithms are amazing human achievements, but they are not good teaching tools because mimicking step-by-step procedures actually traps students into using less sophisticated reasoning than the problems are intended to develop. (unclear).

Kim  00:31

I thought you were going to sneeze!

Pam  00:35

No, I had to take a deep breath. That is the longest sentence ever!

Kim  00:40

In this podcast... 

Pam  00:41

Sorry.

Kim  00:42

We hope to help you teach mathing, building relationships with your students, and grappling with mathematical relationships. 

Pam  00:49

We invite you to join us to make math more laughable and figure-out-able. Bam! Because it's more fun that way. Hey, how's it going? Hey, last week, we started with a lovely paper that one of your kids brought home a while back, and we discussed how there's a lot that's going on in exponents. And really, there's a lot as we go further out, the development of mathematical reasoning in the hierarchy. Every time we get to one of those new ovals, we're dealing with more and more sophistication, and that part of that sophistication is dealing with more things simultaneously. And we need to build schema, so that we have that sort of sort of cinched as a thing that we can then use to operate on other things, and we don't have to refigure everything every time. But unlike some people who say, "Just memorize that thing," we want to be able to, if we need to, get into that schema and actually understand what's happening because that will influence understanding new things. So, we decided that we would do a few podcasts where we could talk about a sequence of tasks that could help kids build a sense of exponentiation, of exponents, and how exponents are multiplicative. We started with one last week. Are you ready for one today to build on that? Sure enough.  Alright, here we go. Do you have a pencil? I do.  Okay.

Kim  02:12

Actually, I'm working off this really shiny... Sorry, I get... I shouldn't use these pencils. So, one of our journey members, we met her... You had met her before, but I met her officially in Chicago. 

Pam  02:24

Oh, yeah (unclear).

Kim  02:27

(unclear). It's Lorna. We met Lorna. 

Pam  02:29

Aww.

Kim  02:29

Who is Scottish and lives in Australia, but she got me these amazing, shiny, optical illusiony looking silver pencils, and I can't stop using.

Pam  02:40

Oh, very nice!

Kim  02:41

(unclear). I feel like my kids who have some focus problems sometimes so...

Pam  02:46

Well, and she also got me a very nice set of pens. But I'll admit, I actually not writing with them today, but I have been. I have them in a different place, and that's where they belong. Yeah. 

Kim  02:56

Alright, I'm ready.

Pam  02:57

Do you take that pencil with you everywhere? Or does it live in like a place?

Kim  03:01

I have a couple of pencils that stay by my computer, and this is one of them. Yeah.

Pam  03:07

Yeah, I'm kind of picky. I have a certain pen in my purse. I have a certain pen in my backpack. I have a certain pen at my desk. Is that weird? It's a little weird.

Kim  03:15

Well, it's weird that you like pens. So...

Pam  03:19

Okay, true that. Alright, let's start with a Problem String that's going to sound maybe a little bit weird in the first problem, but the first problem is literally 4^3. And I'm going to say "four cubed", and I'm going to write a big number 4 with a little 3 raises the exponent. And I'll just ask, what does it mean when mathematicians write this little number 3 called an exponent? What does that mean? And then I'll just... You know, if I'm doing this with real students, I may or may not get correct answers, but we'll talk about a little bit. Now, I'm not going to have them guess what's in my head. If they know nothing, maybe this isn't the best Problem String for them because you should have done something like we did in the last episode, where they have a feel for what that little number means. So, this would be a little bit of a review, if I'm going to ask it that way. So, Kim, little bit of review. What does it mean if I have 4^3?

Kim  04:12

That means you have 4 times 4 times 4.

Pam  04:14

Cool. And any ideas what 4 times 4 times 4 is? 

Kim  04:18

It is 64

Pam  04:19

How do you know?

Kim  04:21

Because I know 4 times 4 is 16. And I know 16 times 4 is 64.

Pam  04:26

Cool. Alright. Question, is there any other way that you could write 64 as an exponent? Like some other number raised to a power?

Kim  04:37

Oh, yeah. 8^2. 

Pam  04:38

Ah, nice. Got anything else?

Kim  04:46

It's a bunch of 2s. How many 2s is that? Six 2s? 2^6.

Pam  04:51

How do you... Why? How do you know? 

Kim  04:53

Because I just wrote 4 times 4 times 4. And each of those 4 has two 2s.

Pam  04:58

Multiplying them. Mmhm. 

Kim  04:59

Mmhm.

Pam  05:00

Yeah. Cool. Excellent. Nicely done. So, on my board right now, I would have 4^3. 4^3 equals 4 times 4 times 4 equals 64 equals 8^2 equals 2^6. 

Kim  05:14

My paper looks exactly like yours. 

Pam  05:16

Ha! Look at that! Alright.

Kim  05:17

That's hilarious. 

Pam  05:18

Totally cool. Next problem. How about, given all that that we just sort of learned, what would 3^3 mean, and then what's it equivalent to?

Kim  05:29

It means 3 times 3 times 3. 

Pam  05:32

Okay, so I've just written that down.

Kim  05:33

Mmhm. And it's equivalent to 27. 

Pam  05:36

And that's because? 

Kim  05:39

9 times 3 is 27.

Pam  05:40

Cool. The third problem is 3 times 3^2. And maybe I should note that for the times symbol I'm using the dot. 

Kim  05:51

Yep.

Pam  05:52

3 times 3^2. Okay, what's 3 times 3^2? 

Kim  05:55

It is also 27.

Pam  05:58

Okay. And how do you know? 

Kim  06:00

Because the 3^2 is 3 times 3. 

Pam  06:03

Okay.

Kim  06:04

So, I wrote 3 times 3 times 3. 

Pam  06:06

Ah, so it's kind of like the problem we had before. So, you kind of use the definition of 3^3 to write 3 times 3, but we had this extra 3 because it's 3 times 3^2. Cool.

Kim  06:16

I have a question. On your paper.

Pam  06:18

Yeah?

Kim  06:18

Did you just write 3 times 3 times 3 with some parentheses or without? 

Pam  06:22

Without. But I could have put them in. 

Kim  06:26

Yeah. 

Pam  06:26

I mean, you asked what I actually did (unclear).

Kim  06:27

Yeah, yeah, yeah.

Pam  06:28

But I can see what you're saying. I could definitely have written 3 times. And maybe that would be a grand thing to do. I might even write it next to it as 3 times 3 times 3, and then put parentheses around the second two because you said you did 3 times 9. Or the 3^2 was 9? 

Kim  06:45

Mmhm.

Pam  06:45

Yeah, fine thing to do. Cool. And then also noting, since it's like the problem before, and we said the problem before, 3 times 3 times 3, was also 3^3, I've also written equals 3^3. And then, I might step back, and I might go, "Huh," so 3 times 3^2 is 3^3. 3^3. 

Kim  07:08

Mmhm.

Pam  07:10

And then I might say, it's almost like there's a 1 for that 3. That first 3. 3 times 3^2. It's almost like that that's one 3. But we just don't usually write that 3^1. And then we have 3^2. And then that's like 3^3. There's sort of three of that. That the little 3, the exponent, means that we had 3 factors of 3. So, I might just stick that 1 above the 3 just for fun. Okay, cool. Next problem. How about 4^2 times 4? Now, usually at this point, somebody's smiling. And I look for somebody who's smiling. I don't know if you were smiling or not, Kim, but what are you thinking about? 4^2? 

Kim  07:54

I was trying to decide if you wanted me to say something or not. 

Pam  07:57

Oh, haha.

Kim  07:58

It is also 64.

Pam  08:00

Okay. 

Kim  08:00

Because 4^2 times 4 is equivalent to 4^3.

Pam  08:06

Cool, so I've just written down 4^2 times 4 equals 64 equals 4^3. I might say, "How did you know that 4^3 was 64." Yeah. Do you have anything?

Kim  08:23

Because I looked before that and saw that I had done three 4s multiplied, and that was 64.

Pam  08:29

Cool. And that was our first problem. 

Kim  08:31

Mmhm.

Pam  08:31

I might look around or ask somebody, "Did anybody think about it as 16?" Because the 4^2? "Did anybody like do the 4^2 as 16, and then think 16 times 4?" And then somebody might say, "Yeah," and so I might write down 16 times 4. And we did that maybe in the first problem. Many, many kids did that in the first problem. So, you're saying that 4^2 times 4 is equivalent to 4^3. I might also then with that 4^2 times 4 put in a little 1 above the 4, so it's like 4^2 times 4^1. And that's 4^3. Okay, cool. Just a few different ways of writing that.

Kim  09:06

Mmhm.

Pam  09:06

At this point, I might say... This might be a time where kids are talking about these numbers. And we found, as we experimented with these Problem Strings with kids, that about this point in the Problem String, kids would be saying things like, "Well, the big 4 to the little 2." Or they wouldn't even say "to the". They would say, "big 4, little 2." And then 4, 3." And they wouldn't really have words necessarily. The same words that I was using. And so, somewhere in these first few problems, we would define, and over on the side of the board, we would say, "Hey, if I wrote something like 4^3, then we're going to call 4 the base." And I would like put the word "base" next to that 4. And then with that third, we would call that a power or an exponent. Let's see. Let me... I'm actually going to call it an exponent. The little guy is called the exponent. And the whole thing, the. 4^3 is called a "power". So, the little number up there is an exponent. The big number is the base. And the whole thing is called a power. And then at that point, I'm just going to kind of raise that as, "Whew! You guys are like wanting to be able to describe these things because you're trying to tell me what to put on the board, so I'll just give you some vocabulary that you can kind of use." And then as they continue from here, I'm not going to wag my finger when they do it wrong. I'm just going to like point to the words, and they're going to go, "Oh yeah, yeah, yeah. That one's the exponent." I'm just going to kind of help them as they're begging for it to be... They want to be able to tell me what to write on the board. I'm just going to kind of help them know, "Oh, yeah. This is what you call it." Oh, yeah, yeah. This is the name for that. This is the way you say that socially." Those would be social things, social convention that we decided as a mathematical education community, or maybe just as a mathematics community, that that's what we were going to call these things, and that's what's going to look like. When we have this kind of multiplication going on, it looks like and we call it that. Cool. Next problem. Let's see. What did we just do? We did 4^2 times 4, Okay. How about next problem? 2^2 times 2^3. 2^2 times 2^3.

Kim  11:13

Mmhm. 

Pam  11:14

What's that equivalent to? 

Kim  11:16

32.

Pam  11:17

Okay. And how do you know?

Kim  11:22

Because way at the top, we wrote 2^6. 

Pam  11:28

Way at the top.

Kim  11:29

The very first problem was 4^3, so that was 64.

Pam  11:32

Okay? 

Kim  11:32

And we said that that was equivalent to 2^6.

Pam  11:35

Ah! Alright, can I slow you down a little bit? 

Kim  11:37

Sure.

Pam  11:38

I love how you actually did it. Let's... I want that strategy in a minute. And so, I'll just share that if this is happening in class, and I have Kim, maybe I don't call on Kim first. But also, if I do, and Kim starts sharing something that's just a little bit more advanced, I might say, "Did anybody think about the 2^2 first?"

Kim  12:01

Mmhm.

Pam  12:02

And so, Kim, if I said that, can you run with that? Or do you want me to just say what I think a kid would say?

Kim  12:07

2^2 is 4. 

Pam  12:08

Okay. And then what's 2^3?

Kim  12:11

That is going to be 8.

Pam  12:12

So, now I've written 2^2 times 2^3 equals 32 equals 4 times 8. Oh, yeah. Sure enough. That's 32. Cool. Then I might say... Let's go with your strategy now. So, yeah, go ahead.

Kim  12:29

Would we also then write that's equivalent to 2^5? (unclear).

Pam  12:35

Well, that's why I was pausing because I wasn't...

Kim  12:37

Yeah.

Pam  12:37

I was not sure if I was going to go with your strategy next or... Would you go with 2^5?

Kim  12:41

I think I'd write 2^5. And then I might go back to me and say, "And how does that relate to the one you just were talking about?" But only if it's important to the string, right? You might come after me... Or come after me. Come to me at the end, right afterwards, and like have a private conversation to like pat me on the back or something. 

Pam  12:58

Sure. But I'm going to tell you, before I wrote 2^5, though, I actually think I would say something like, "Okay, so you guys said 2^2 was 4 and 2^3 was 8. 2^2 how many 2s?" And then I would write 2 times 2. 

Kim  13:11

Mmhm.

Pam  13:11

"And 2^3 is how many 2s?" And so, next to that 2 times 2, I would write times 2 times 2 times 2. So, now I have 2 times 2 times 2 times 2 times 2. And I would say, "How many 2s is that that we're multiplying? And when they say 5, then I would write, "How do we write that? How do we..." Or I would say, "How do we write that?" And they would say, "Well, the base is 2, and the exponent is 5." And that's how I would get to that 2^5. 

Kim  13:39

Yep.

Pam  13:40

Yeah. And so, I would say, "Oh, so you're saying that 2^2 times 2^3 is 2^5. Everybody agree? Disagree?" Kim, go ahead and share what you were going to say, even though I might... Nah, I think I might have. I might do it now.

Kim  13:53

We had just said that 2^6 was 64.

Pam  13:56

Mmhm.

Kim  13:56

And so, this is going to be 2^5, so it's half as much because I have one less 2 being multiplied.

Pam  14:06

One less 2 being multiplied, so it's going to be half as much as the 64, which then makes this 32. Nice. Everybody out there who's thinking, "Pam, just teach kids the rules and they'll totally be good." I would suggest that what you just heard Kim do shows that she actually is thinking and reasoning about exponents, and that she's not just applying rules. The fact that she can look at 2^5 as being half of 2^6 is huge. I can't tell you... Yeah, go ahead. 

Kim  14:36

Can I tell you though, that when I look at 2^2 times 2^3 for just a second, I'm like, "Am I supposed to add those or multiply them?" Like, I don't remember. Outside of thinking about what that means, I literally don't remember if there's a rule what it is. I mean, I'm sure there are rules people teach. I would look at that, and I have to like think about what it is, and then I'm confident.

Pam  15:04

Yeah. And, listeners, I wonder how you feel about that. We'd love to hear. Send us some comments. How do you feel about the fact that Kim, the successful mathematician, when she looks at 2^2 times 2^3, says to herself, "I think there's a rule that goes there, but I don't know what it is, so I'm just going to think about it." And and I will also suggest that my kids do the same thing. They don't... It's not some weird rule. Now, let's keep going because there might be... Yeah. So, we'll raise that idea. And I'm glad you said it. Nicely said. Let me think about where we are. Find myself in the string, Pam. Okay. Next problem. What about 2^4 times 2^2? What do you got?

Kim  15:50

That is 2^6, so it's 64.

Pam  15:53

How do you know? Because you didn't just add the 4 and the 2. 

Kim  15:58

Well, I know it's four 2s multiplied and two 2s multiplied. 

Pam  16:01

Okay, so as you say that, I'm writing on my paper you know that 2^4 is 2 times 2 times 2 times 2.

Kim  16:07

Mmhm.

Pam  16:07

And then what? 

Kim  16:09

2^2 is two more 2s multiplied. 

Pam  16:11

So, I just wrote that down. So, right now I have six 2s with multiplication symbols in between them.

Kim  16:16

Mmhm. 

Pam  16:17

And that's... Oh, and then I wrote equals 2^6. Okay, cool. 

Kim  16:21

That is 64.

Pam  16:22

And how do you know that? Oh, because we did it before.

Kim  16:24

Mmhm.

Pam  16:25

Yeah, okay. Cool. Nice. So, it almost seems like 2^2 times 2^3 was 2^5. And 2^4 times 2^2 was 2^6. I wonder if we're seeing some patterns happen here. There's one more pattern I was looking for. Oh. When you said that that was also 64, I might also write that equivalent to 4^3 just from that first problem. 

Kim  16:52

Mmhm.

Pam  16:53

So, now we've got just sort of lots of exponent relationships happening. So, based on kind of everything that we've just done. If I just... I know this is kind of weird. This next problem in the string is a little strange. What if I just said, so what is any number? I'll just write x. So, I've just written an x. And I raise it to some random exponent. And I'll just write m. So, I've written x^m. What does that mean? If you just see some random number, x, raised to some random other exponent, m, what could that say to your heart and your soul? And then I would let kids talk about that for a minute. So, listeners, I wonder what you think kids might say at this point. And, Kim, do you want to just tell us what's going on in your head? 

Kim  17:41

Yeah.

Pam  17:43

Your chuckling. Yes?

Kim  17:44

I'm off task. 

Pam  17:49

Okay. 

Kim  17:49

I'm currently obsessed with the 2^6 and the 4^3, and I want to play with some other numbers. (unclear)

Pam  17:54

Ah, well that's kind of fun.. Alright. So, if you want to play with that, I'll tell you (unclear)...

Kim  17:58

No, it's okay. So, this is x multiplied m number of times. So... 

Pam  18:03

Yeah. And so, I'm just kind of curious. Like, listeners, how would you represent that x multiplied... Say that again? How'd you say that?

Kim  18:10

X multiplied m number of times.

Pam  18:14

M number of times? I'm actually going to push back on that just a little bit because I think it's m, x's. So, it's just "the number of times" is a little tricky because the number of multiplications is actually less than the number of x's because...

Kim  18:31

But we've been saying 2 two times or 2 three times, multiplied 3 times. How is that different?

Pam  18:38

So, like let's go back to 4^3. The very first problem that we did.

Kim  18:41

Mmhm. 

Pam  18:41

So, there's three 4s. Agreed? But there's two multiplications.

Kim  18:48

But didn't I say 4 multiplied 3 times?

Pam  18:52

I don't know. But I think three 4s. There's three 4s being multiplied.

Kim  18:59

Yeah, I guess I just don't hear how that's different (unclear).

Pam  19:01

How that's different. 

Kim  19:01

(unclear) 4.

Pam  19:02

So, 4 times 4 times 4. How many multiplications is that?

Kim  19:08

Oh, you're saying the action of multiplication is only two times.

Pam  19:12

Yeah, yeah. 

Kim  19:12

Okay.

Pam  19:13

Yeah. So, love it. So, if I were doing x^m, I would say, so, that's like, x times x times x times dot, dot, dot, dot times x. And how many x's are there? M. There's m, x's? Yeah, yeah. I only say that because I wrote exactly what you said, and then we had a bunch of reviewers, and they were like, "No, no, you can't talk about that number of multiplications." And I was like, "What?" And it took me much longer than it just took you to think about, it's that many factors. It's m factors of x's. M. Am I saying that right? M? M, x's, M, x's. Yeah, cool. Cool. So, if we have that generalization that some number raised to a power is I'm going to multiply that number. I'm going to multiply m, x's, then we could also maybe give some kind of crazy problem. Like, what is 54^15 times 54^3? And I just want the answer in terms of 54s.

Kim  20:17

Yeah, I'm not solving that. That's 54^18. 

Pam  20:21

And how do you know? 

Kim  20:23

Because you have fifteen 54s and three 54s.

Pam  20:29

Multiplied together.

Kim  20:30

Mmhm. 

Pam  20:30

And so, then you would have eighteen 54s multiplied together.

Kim  20:32

Mmhm. 

Pam  20:33

Yeah, super. And then I would be done with that Problem String. And the goal of that Problem String... Now, it's funny because if anybody was writing any of this down, once we got to... We had 2^2 times 2^3 and 2^4 times 2^2. I think a lot of high school or algebra teachers will be screaming at this point, "Sweet! We've got the sort of product rule, or the addition rule," or however you want to call that. And I'm going to push back a little bit. It's a relationship. But at that point, I think we can clearly say that what we've got going on is we're really dealing. Yes, we could have gone there, but I'm not going to go there yet. I'm going to give students an experience to really get at what does x to the m mean? What does it mean to have a base raised to an exponent and really hammer that down. Not just getting kids to start spitting out a rule. And then the goal was the rule. And the goal really wasn't understanding. And why didn't you just give us the rule to begin with? Because we're not going for rules. We're going for understanding. 

Kim  21:30

Well, yeah, because I can tell you in this moment that you're adding the exponents because I'm like right here doing it. But in three years, if I never made sense of what was happening, I would not be able to regurgitate that, and I would not have any clue how to attack this problem. And like that's what we have with kids who memorize something in one moment, and then they are not involved in on a regular basis. We pull it back for review, and all of a sudden it's not review because they don't have anything to attach. They didn't attach the learning to anything. They just try to memorize something. 

Pam  22:03

Sure. And if you're a super good memorizer, maybe you just memorize it again and you carry on. But I'm just going to submit that's not mathing. Like, if we can actually understand what's going on, then exponents can make sense, and we are reasoning exponentially. Which is sort of what we're going for here. Alright, Kim, that was kind of fun. 

Kim  22:22

Yeah.

Pam  22:23

Thanks for doing that string. And, ya'll, thanks for tuning in and teaching more and more real math. Tune in next week to develop more exponential relationships. To find out more about the Math is Figure-Out-Able movement visit mathisfigureoutable.com and keep spreading the word that Math is Figure-Out-Able!